L8a10

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L8a9.gif

L8a9

L8a11.gif

L8a11

Contents

L8a10.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8a10 at Knotilus!

L8a10 is 8^2_{5} in the Rolfsen table of links.


Symmetric version
Mongolian ornament

Link Presentations

[edit Notes on L8a10's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X12,15,13,16 X14,5,15,6 X4,13,5,14 X16,11,7,12 X2738 X6,9,1,10
Gauss code {1, -7, 2, -5, 4, -8}, {7, -1, 8, -2, 6, -3, 5, -4, 3, -6}
A Braid Representative
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A Morse Link Presentation L8a10 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(2) t(1)^2-t(1)^2+2 t(2)^2 t(1)-3 t(2) t(1)+2 t(1)-t(2)^2+2 t(2)}{t(1) t(2)} (db)
Jones polynomial -\frac{1}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z-a^7 z^3+a^7 z^{-1} -2 a^5 z^3-3 a^5 z-a^5 z^{-1} -a^3 z^3-a^3 z (db)
Kauffman polynomial -z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-2 z^6 a^{10}+6 z^4 a^{10}-4 z^2 a^{10}-z^7 a^9+4 z^3 a^9-z a^9-4 z^6 a^8+8 z^4 a^8-3 z^2 a^8-z^7 a^7-2 z^5 a^7+6 z^3 a^7-4 z a^7+a^7 z^{-1} -2 z^6 a^6+2 z^2 a^6-a^6-3 z^5 a^5+4 z^3 a^5-4 z a^5+a^5 z^{-1} -2 z^4 a^4+z^2 a^4-z^3 a^3+z a^3 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-8-7-6-5-4-3-2-10χ
-2        11
-4       21-1
-6      2  2
-8     22  0
-10    32   1
-12   12    1
-14  23     -1
-16 12      1
-18 1       -1
-201        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z} {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L8a9

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L8a11